June 20, 1913] 



SCIENCE 



945 



lowing tilings happen to be facts, we can not 

 ever hope to detect the absolute motion of the 

 table by experiments on light or electricity. 



1. If there is a bar AB on the table point- 

 ing in the line of the motion, to the stationary 

 observer its length would be only AB\/(1 — ^) 

 where ^ =■ v/c. 



2. The time on a clock at A on the table 

 would read less in the same ratio, that is, if the 

 motion began at noon, when the stationary ob- 

 server knew that the time was really t the 



clock would read iV(l — iS^)- 



3. A clock at B could not be made to read 

 the same as a clock at A at the same instant 

 but would be behind that at A by 

 AB y _ 



If a clock were instantly moved from A to B 

 the hands would instantly shift through that 

 amount. This is the principle of local time. 



The stationary observer would deduce at 

 once some very startling conclusions, such as 

 these. If the table could move with the veloc- 

 ity of light, ;8^1, and the length of AB 

 would be nothing at all. The clock at A 

 would cease to register time at all. The ob- 

 vious conclusion would be that the velocity of 

 light is a maximum that no velocity could 

 ever reach. But even for velocities below that 

 of light we have to give up the idea of incom- 

 pressible bodies. Energy and mass become 

 confused and physics has to be remade. And 

 the difficulty of time being attached to the 

 place at which we are, so that no time meter 

 could be devised which could be moved around 

 and retain its correct reading, is disturbing. 

 If two clock faces are at the ends of a long 

 axis, and read together when across the line 

 of motion, why should there be a twist in the 

 axis when it is turned into the line of motion? 



To enable one to understand these proposed 

 relations of distance and time, Minkowski 

 conceived the notion of giving them a geo- 

 metrical setting. This is nothing new in 

 physics, for many models have been made to 

 represent various laws and hypotheses. They 

 enable us to look at the relations in a much 

 more direct way; to be able, as it were, to look 



over a map of the ground. It must be borne 

 in mind, however, that such representations 

 are not substitutions for the thing itself. A 

 temperature-entropy diagram is not steam in 

 a boiler, of course, but only shows certain re- 

 lations as to the steam in the boiler. So too, 

 Minkowski's geometric setting of relativity is 

 not a picture of the world, but a representa- 

 tion of the relations that are set forth in the 

 theory of relativity. 



His suggestion was that if we use a four- 

 dimensional space, measuring x, y, z (which 

 give us the position of the laboratory table) 

 along three axes, and measure on the fourth 

 axis the distance ct, then the fourth axis can 

 be spoken of as a time axis, since c is a con- 

 stant. In this way we can speak of the situa- 

 tion of the real world at time i as a section 

 in the four-dimensional world by moving a 

 space of three dimensions. The idea is easily 

 illustrated by imagining a wave on a pond 

 made by a stone dropped into the water. The 

 wave spreads out with a given velocity. If 

 now we construct a cone of the proper angle, 

 immerse the point at the center of the wave 

 and let the cone sink at the right speed, the 

 expanding wave will always remain in contact 

 with the cone. Or, so far as geometry is con- 

 cerned, we can keep the cone stationary and 

 let a cutting plane move upward. The circu- 

 lar section on the plane will then appear to 

 expand like a wave. In an analogous manner 

 we can at least get a phraseology that will de- 

 scribe the ideas underlying relativity of the 

 electrodynamic kind. It turns out that if we 

 represent these in a four-dimensional space 

 the whole statement of the relativity property 

 can be summed up in one simple statement, 

 that is : In the four-dimensional space the 

 choice of our axes of reference is fairly arbi- 

 trary. We may take axes inclined at the 

 proper angle to our original axes, as new axes 

 of reference, and the equations for the new 

 x' , y' , z', V, are just like the original equa- 

 tions. Indeed if we suppose the table men- 

 tioned above to move along the x axis, as 

 viewed by a stationary observer, with a uni- 

 form velocity v, which we may set equal to 

 c tanh <^, where tanh (f,^ p, tanh' being the 



